Best question of the semester

Quick break from prepping and grading final exams.

My future elementary teachers always struggle to name the denominator when they need to find \frac{1}{4} of 1\frac{1}{2}.

They draw the picture.


They know that the numerator needs to be 3. And then they argue about whether the denominator should be 12 or 16.

I struggle every year to get 8 on the table as an acceptable answer. I usually end up being a voice of authority for 8, and we discuss what the whole is if you use 12 or 16 as the denominator.

My students don’t like 8 because that means the answer is \frac{3}{8} of one square, but the pieces come from different squares.

This year, I had an insight that helped a lot. The question was this:

What are some situations in life when you get two same-sized parts of distinct wholes?

I opened the class session following our usual denominator debate with this question and it helped us to focus on the issue at hand.

After a few false starts (i.e. examples that didn’t really exemplify what we were after), we settled on this scenario.

When you buy a 75¢ pop from a vending machine, by inserting a dollar you get back a quarter. Do it again and now you have two quarters. Each quarter came from a different dollar, but they are still quarters. Each is one-fourth of a dollar and together they are half a dollar (even though collectively, they are one-fourth of the money you started with).

Back to the squares and we had a frame of reference for eighths.

I have been teaching this course for 8 years.


15 responses to “Best question of the semester

  1. >I have been teaching this course for 8 years.

    Sounds like you think you should have seen this before. I am still learning things in every class I teach. (It’s wonderful to see the mathematics become more and more densely connected, and it’s scary to think how differently my students see it.)

  2. Michael Paul Goldenberg

    Interestingly (I hope), as soon as I saw this, I started thinking about the numerator I’d want so that I could divide it by 4 (same as multiplying by 1/4). Multiplying the other fraction by 2/2 (having changed 1 1/2 to 3/2) gave me 6/4. No good, as 6 isn’t divisible by 4. Multiplying again by 2/2 gave 12/8. Bingo. 12/4 = 3 => the answer is 3/8. i just let the denominator “take care of itself.”

    I took this approach because I was reading the problem on-line and didn’t want to be bothered with getting pencil and paper. Mental arithmetic is, to my mind, a huge advantage, and not worrying at all about finding the denominator before I conveniently “ran into it” made this a snap. Just sayin’.

  3. What do they think one of the shaded blue rectangles stands for, in terms of its quantity? If they can see that they have 3 measuring units of size 1/8 (of 1 unit), then 3/8 makes sense, so that’s why I ask whether they can see a single shaded blue rectangle as 1/8.

    I think the conceptual hurdle that you identified (pieces coming from different squares) is interesting!

  4. Seeing ‘pop’ makes me happy. It took me 10 years away from MN to get used to saying soda.

    Interesting (almost strange) that they don’t like 8 as denominator when the drawing SHOWS that each blue is 1/8, especially when they are okay with 3 as numerator. (Sorry, this is what Mandy just said.) I’m leaving now.

  5. It’s neat that there is a lived experience of this that you were able to come up with and connect to. I can see why 3/12 is a compelling answer when looking at that diagram (I’m not convinced by the 3/16 camp, sorry guys) and the connection to the soda machine story helps!

    One thing I’m puzzled over (and it may be that just writing it out will unpuzzle me, or that someone will say something that makes me say “duh!”) is what kinds of operations on fractions shift the whole, and what kinds don’t.

    Multiplication: it seems to me that 1/4 * 3/2 = 3/8, the whole is in reference to whatever whole the 3/2 means. So if the 3/2 is 3/2 of a carton of eggs (e.g. 18 eggs) then 1/4 of that is 4.5 eggs, and 4.5 eggs is 3/12 of the total 18 eggs (of course — we’re aiming for 1/4 of that total) but it’s 3/8 of the original whole (a dozen eggs). The whole doesn’t change.

    Division: the model of division that makes the most sense to me is “how many of these are in those” — like, how many 1/4 of a yard are in 3/2 of a yard, for example. The answer is 6, but 6 whats? Definitely not 6 yards! The answer is 6 1/4s of a yard — the unit shifted! Even if you wanna do eggs, you could say you have one and a half dozens of eggs, and you want to know how many 1/4 of a dozen eggs are in it. There’s 6 — 18 eggs / 3 eggs = 6 (sets of 3 eggs). The answer’s not 1/2 (6 eggs is half the original unit of a dozen eggs) — the egg units cancel and you have this unitless ratio, kinda.

    Addition: 1/4 + 3/2, if you’re thinking about dozens of eggs again, that’s 3 eggs + 18 eggs = 21 eggs, or 21/12 of a dozen (aka 7/4, the “right answer”). The units are still a carton of eggs.

    Subtraction: 3/2 – 1/4, 18 eggs – 3 eggs = 15 eggs, and 15/12 of a dozen.

    Why is division so weird, or maybe, why isn’t multiplication so weird? Is part of why we lose our intuition around units and fractions because multiplication and division behave so differently?

    Finally, a question that’s been bouncing around The Math Forum for a while is: “is it fair to say that the numerator of a fraction tells how many ‘pieces’ and the denominator tells what size ‘pieces’ the whole is broken into?” Would the “how many?” “what size part of the whole?” questions help students see that each piece is 1/8 of the whole and we have 3 pieces?

  6. @Sue and Christopher – I really liked reading “I have been teaching this course for 8 years.” I haven’t been teaching anything for 8 years and it’s reassuring to see the kind of command of detail that can develop over time (and, it seems, only over time).

    @Fawn and Christopher – I’m from MN too! Yay for pop!

  7. Wow. You all are way more interested in this than I expected anybody to be. Lots of food for thought and discussion here. I’ll respond briefly here in the comments to your ideas, and then there is probably a full-blown blog post for some meta-analysis coming soon.

    Sue and Dan: My intention with the I have been teaching this course for 8 years comment is exactly what Dan mentions. I am not beating myself up; I am noticing how incredibly rich and challenging this job of teaching is. It took me several years to notice the challenge, and then several more years of mulling it over before I was able to develop this one question. It’s not going to cure the problem, but it changes the nature of our conversation in a productive way.

    MPG: You are pointing to another important thing-mental math strategies. I need to work harder on those with these students. I have some ideas, and I have made some strides. But I need to do more. Thank you for the reminder.

    That said, the goal of the activity I have described isn’t just to find the right answer (with any route to that answer being acceptable). Important goals (which were unstated in the post) are to get better at (1) using diagrams to solve problems, and (2) interpreting the whole when we’re dealing with fractions. Your mental math solution downplays both of these issues.

    Mandy and Fawn: This is a lovely question. We do discuss this, and the class is usually cool with saying that one blue piece is 1/8 of a square. But (some protest) the eighths come from different squares. So you don’t have 3/8 of a square. You have 2/8 of one square and 1/8 of another square. Or 3/16 of the squares.

    And here’s the problem…that last fraction is not wrong. Not wrong at all (even if Max isn’t convinced by it). With a correctly identified whole, 3/16 is a correct answer. So is 3/12.

    There is a brilliant IMAP video in which a girl solves the problem, If 5 children share 2 peanut butter cookies equally, how much does each child get? After much work, she says that each child gets 1/5 of each cookie or 2/10 of all the cookies. Much questioning by the interviewer fails to get her to state 2/5 of a cookie as a possibility. That is what my students are wrestling with.

    Finally, Max, you are working on the very thing we have been dissecting in class. What is one? (Regular readers will not be surprised to find this theme surfacing on this blog). You note that units change when we multiply fractions. This is true, but it’s a characteristic of multiplication more generally. 2×3 typically means 2 groups of 3. The 2 and the 3 refer to different units; they count different wholes.

    I’ll have to invest some time thinking about your Math Forum question.

    • Michael Paul Goldenberg

      Nice responses to great comments, Christopher. One of the things I would do myself had I truly not felt that I understood the underlying math more conceptually would be to reexamine it for “understanding” after doing my calculation. That can be a hard sell to students of any age, however, particularly to teachers and would-be teachers, in my experience.

      One of the biggest obstacles I’ve had in doing professional develop on mathematics with elementary school teachers is the sort of teacher who says, “Why do I need to go through all this abstraction you’re laying on us? I got the right answer, didn’t I?”

      Of course, that’s a symptom of something deeply wrong in how mathematics is generally taught in the US. Sadly, we then have people who learned that particularly lesson all too well instructing the next generation. And the bleat goes on.

  8. Christopher, I agree that 3/16 isn’t wrong. It’s just not the one I’d pick, personally. I find the argument for that less compelling than the arguments for 3/8 (three of the eight pieces of the whole) and 3/12 (three of the twelve pieces we started with), though it’s not like I think the 3/16 (three of the sixteen pieces in the two wholes that our twelve pieces came out of) camp is being less thoughtful.

    Here’s the thing that’s bugging me about the units changing… The units might be changing (from halves and fourths to eighths), but the *whole* isn’t changing in multiplication, only in division. In multiplication, I can multiply hours and miles per hour and get miles, or even kilowatts and hours and get kilowatt-hours, but I can’t add miles and hours or kilowatts and hours. There’s an analogy often made to fractions, that I can’t add halves and fourths so I must change both to eighths. But is that the same kind of changing of units as multiplying halves and fourths and getting eighths?

    And is changing units different than changing wholes? Why does dividing feel so dramatically different, while multiplication, addition, and subtraction feel the same?

    I’m excited to read the meta-post or whatever you cook up out of this conversation, and to keep playing along. I do love fractions!

  9. Do they have the same sort of issues when you use a number line model? Interesting idea about the ambiguity of one in the area models. I am never quite comfortable with area models – maybe that is part of the reason.

  10. Max, In the division problem we are calculating “how many” which is a unitless number. The question is of the form: _______ 1/4 yards are in 3/2 of a yard. The units for the answer, 6, are already provided in the phrasing of the question. Not so with the multiplication problem: 1/4 of 3/2 of a carton of eggs is __________. I am having trouble thinking of a way to phrase a dvision problem that requests a value with units.

  11. Interesting idea here. I suppose your point, hodge, is about the linguistic form of the question, rather than the nature of 6 (which does in fact have a unit: 6 what? 6 quarter-yards). Do I have that right? And so you are trying to think of a division question that doesn’t provide the units ready-made in the question itself?

    I’ll have to ponder this for a bit. I’m not sure whether I believe that it matters. But I can’t produce an argument that it doesn’t either.

    And I’ll also have to think further about Max‘s assertion about differences between wholes and units, which I have come to think of as synonymous.

  12. This units business is quite confusing. But, I cannot see any difference between the mathematical and linguistic treatment of units. Both of these equations result in the same numerical division. Mathematically, one solution has units and the other does not.

    a) (1/4 yard) x = 3/2 yards b) ¼ x = 3/2 yards
    Linguistically the problems might be: a) how many ¼ yards in 3/2 yards and b) How long is a rope if a quarter of the rope is 3/2 yards. Linguistically, one calls for an answer with units and the other does not. “How many” does not call for an answer with units. “How long” does call for an answer with units.

    I am honestly curious whether you get the same issue about a denominator of 12 or 16 when you use a number line model for the original question. Or do you not typically use a number line model with your future elementary school teachers?

  13. Ah, right. I do owe you a response on the number line thing, hodge. Sorry.

    I suspect that you are right; a number line would not create the same issues. I haven’t spent time with number lines and multiplication, so I can’t say what the entailments would be. But I think you are right that this denominator issue wouldn’t surface.

    I have a close colleague here in MN who swears by multiple representations-doing every operation with a bunch of different models. There is wisdom in that, I think. I shall endeavor to incorporate more multiple representations work into the course. Surely, I will document successes and failures of doing so in this space.

    But I still want to have the conversation about correct ways to interpret that area model above. If students can see it as 3/8, 3/12 and 3/16 all simultaneously and with correct units attached, then we have achieved an important mathematical objective.

    On the units issue, two things. First, do I summarize you correctly thusly, “Counting is a unitless activity”? Perhaps you have seen the video recounting my own views on this matter?

    Second: Your question (b) could be rephrased as How many yards long is a rope if a quarter of the rope is 3/2 yards?. Now it’s a how many question, but the change feels linguistic to me, not mathematical.

  14. Yes, I agree the area model is a worthwhile use of time and I like the soda machine example. I was not suggesting the number line model as a way of avoiding the area model issues – those issues should be confronted. I was just curious whether the same issue would come up.

    In my mind, numbers are unit-less. I have seen your video and don’t disagree with it, other than I have a different idea of what makes a unit. A dozen is not a unit in my mind. It is a quantity. Same for 10.

    I believe your rephrasing of my question b) is no longer consistent with the original equation. This is clear, I think, if the variable is defined.

    How long is a rope if a quarter of the rope is 3/2 yds?
    x: length of the rope
    1/4 x = 3/2 yds
    6 yds or 18 ft are both valid answers

    Your Version:
    How many yards long is a rope if a quarter of the rope is 3/2 yards?
    x: length of the original rope in yards
    (1/4) (x yds) = 3/2 yds
    note that 6 is the only acceptable answer

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